Let K be a proper cone in R^x,let A be an n×n real matrix that satisfies AK(?)K,letb be a given vector of K,and let λbe a given positive real number.The following two lin-ear equations are considered in this pap...Let K be a proper cone in R^x,let A be an n×n real matrix that satisfies AK(?)K,letb be a given vector of K,and let λbe a given positive real number.The following two lin-ear equations are considered in this paper:(i)(λⅠ_n-A)x=b,x∈K,and(ii)(A-λⅠ_n)x=b,x∈K.We obtain several equivalent conditions for the solvability of the first equation.展开更多
In this paper,we study the multivariate linear equations with arbitrary positive integral coefficients.Under the Generalized Riemann Hypothesis,we obtained the asymptotic formula for the linear equations with more tha...In this paper,we study the multivariate linear equations with arbitrary positive integral coefficients.Under the Generalized Riemann Hypothesis,we obtained the asymptotic formula for the linear equations with more than five prime variables.This asymptotic formula is composed of three parts,that is,the first main term,the explicit second main term and the error term.Among them,the first main term is similar with the former one,the explicit second main term is relative to the non-trivial zeros of Dirichlet L-functions,and our error term improves the former one.展开更多
In this paper, we consider solving dense linear equations on Dawning1000 byusing matrix partitioning technique. Based on this partitioning of matrix, we give aparallel block LU decomposition method. The efficiency of ...In this paper, we consider solving dense linear equations on Dawning1000 byusing matrix partitioning technique. Based on this partitioning of matrix, we give aparallel block LU decomposition method. The efficiency of solving linear equationsby different ways is analysed. The numerical results are given on Dawning1000.By running our parallel program, the best speed up on 32 processors is over 25.展开更多
In this article, we will derive local elliptic type gradient estimates for positive solutions of linear parabolic equations(?-?/(?t))u(x, t) + h(x,t)u(x,t) = 0 and nonlinear parabolic equations(?-?-/(?t))u(x,t) + h(x,...In this article, we will derive local elliptic type gradient estimates for positive solutions of linear parabolic equations(?-?/(?t))u(x, t) + h(x,t)u(x,t) = 0 and nonlinear parabolic equations(?-?-/(?t))u(x,t) + h(x, t)u^p(x,t) = 0(p > 1) on Riemannian manifolds.As applications, we obtain some theorems of Liouville type for positive ancient solutions of such equations. Our results generalize that of Souplet-Zhang([1], Bull. London Math. Soc.38(2006), 1045-1053) and the author([2], Nonlinear Anal. 74(2011), 5141-5146).展开更多
Based on linear interval equations, an accurate interval finite element method for solving structural static problems with uncertain parameters in terms of optimization is discussed.On the premise of ensuring the cons...Based on linear interval equations, an accurate interval finite element method for solving structural static problems with uncertain parameters in terms of optimization is discussed.On the premise of ensuring the consistency of solution sets, the original interval equations are equivalently transformed into some deterministic inequations.On this basis, calculating the structural displacement response with interval parameters is predigested to a number of deterministic linear optimization problems.The results are proved to be accurate to the interval governing equations.Finally, a numerical example is given to demonstrate the feasibility and efficiency of the proposed method.展开更多
This paper deals with the Hyers-Ulam stability of the nonhomogeneous linear dynamic equation x~?(t)-ax(t) = f(t), where a ∈ R^+. The main results can be regarded as a supplement of the stability results of the corres...This paper deals with the Hyers-Ulam stability of the nonhomogeneous linear dynamic equation x~?(t)-ax(t) = f(t), where a ∈ R^+. The main results can be regarded as a supplement of the stability results of the corresponding homogeneous linear dynamic equation obtained by Anderson and Onitsuka(Anderson D R, Onitsuka M. Hyers-Ulam stability of first-order homogeneous linear dynamic equations on time scales. Demonstratio Math., 2018, 51: 198–210).展开更多
Integral equations theoretical parts and applications have been studied and investigated in previous works. In this work, results on investigations of the uniqueness of the Fredholm-Stiltjes linear integral equations ...Integral equations theoretical parts and applications have been studied and investigated in previous works. In this work, results on investigations of the uniqueness of the Fredholm-Stiltjes linear integral equations solutions of the third kind were considered. Volterra integral equations of the first and third kind with smooth kernels were studied, and proof of the existence of a multiparameter family of solutions is described. Additionally, linear Fredholm integral equations of the first kind were investigated, for which Lavrent’ev regularizing operators were constructed.展开更多
In this work,we study the linearized Landau equation with soft potentials and show that the smooth solution to the Cauchy problem with initial datum in L^(2)(ℝ^(3))enjoys an analytic regularization effect,and that the...In this work,we study the linearized Landau equation with soft potentials and show that the smooth solution to the Cauchy problem with initial datum in L^(2)(ℝ^(3))enjoys an analytic regularization effect,and that the evolution of the analytic radius is the same as the heat equations.展开更多
Fundamental matrix operations and solving linear systems of equations are ubiquitous in scientific investigations.Using the‘sender-receiver’model,we propose quantum algorithms for matrix operations such as matrix-ve...Fundamental matrix operations and solving linear systems of equations are ubiquitous in scientific investigations.Using the‘sender-receiver’model,we propose quantum algorithms for matrix operations such as matrix-vector product,matrix-matrix product,the sum of two matrices,and the calculation of determinant and inverse matrix.We encode the matrix entries into the probability amplitudes of the pure initial states of senders.After applying proper unitary transformation to the complete quantum system,the desired result can be found in certain blocks of the receiver’s density matrix.These quantum protocols can be used as subroutines in other quantum schemes.Furthermore,we present an alternative quantum algorithm for solving linear systems of equations.展开更多
Hessian matrices are square matrices consisting of all possible combinations of second partial derivatives of a scalar-valued initial function. As such, Hessian matrices may be treated as elementary matrix systems of ...Hessian matrices are square matrices consisting of all possible combinations of second partial derivatives of a scalar-valued initial function. As such, Hessian matrices may be treated as elementary matrix systems of linear second-order partial differential equations. This paper discusses the Hessian and its applications in optimization, and then proceeds to introduce and derive the notion of the Jaffa Transform, a new linear operator that directly maps a Hessian square matrix space to the initial corresponding scalar field in nth dimensional Euclidean space. The Jaffa Transform is examined, including the properties of the operator, the transform of notable matrices, and the existence of an inverse Jaffa Transform, which is, by definition, the Hessian matrix operator. The Laplace equation is then noted and investigated, particularly, the relation of the Laplace equation to Poisson’s equation, and the theoretical applications and correlations of harmonic functions to Hessian matrices. The paper concludes by introducing and explicating the Jaffa Theorem, a principle that declares the existence of harmonic Jaffa Transforms, which are, essentially, Jaffa Transform solutions to the Laplace partial differential equation.展开更多
In this paper, we mainly investigate entire solutions of complex differential equations with coefficients involving exponential functions, and obtain the dynamical properties of the solutions, their derivatives and pr...In this paper, we mainly investigate entire solutions of complex differential equations with coefficients involving exponential functions, and obtain the dynamical properties of the solutions, their derivatives and primitives. With some conditions on coefficients, for the solutions, their derivatives and their primitives, we consider the common limiting directions of Julia set and the existence of Baker wandering domain.展开更多
It is widely accepted that a robust and efficient method to compute the linear spatial amplified rate ought to be developed in three-dimensional(3D) boundary layers to predict the transition with the e^N method,especi...It is widely accepted that a robust and efficient method to compute the linear spatial amplified rate ought to be developed in three-dimensional(3D) boundary layers to predict the transition with the e^N method,especially when the boundary layer varies significantly in the spanwise direction.The 3D-linear parabolized stability equation(3DLPSE) approach,a 3D extension of the two-dimensional LPSE(2D-LPSE),is developed with a plane-marching procedure for investigating the instability of a 3D boundary layer with a significant spanwise variation.The method is suitable for a full Mach number region,and is validated by computing the unstable modes in 2D and 3D boundary layers,in both global and local instability problems.The predictions are in better agreement with the ones of the direct numerical simulation(DNS) rather than a 2D-eigenvalue problem(EVP) procedure.These results suggest that the plane-marching 3D-LPSE approach is a robust,efficient,and accurate choice for the local and global instability analysis in 2D and 3D boundary layers for all free-stream Mach numbers.展开更多
In this paper,we study the hyperstability for the general linear equation f(ax+by)=Af(x)+Bf(y)in the setting of complete quasi-2-Banach spaces.We first extend the main fixed point result of Brzdek and Ciepliński(Acta...In this paper,we study the hyperstability for the general linear equation f(ax+by)=Af(x)+Bf(y)in the setting of complete quasi-2-Banach spaces.We first extend the main fixed point result of Brzdek and Ciepliński(Acta Mathematica Scientia,2018,38 B(2):377-390)to quasi-2-Banach spaces by defining an equivalent quasi-2-Banach space.Then we use this result to generalize the main results on the hyperstability for the general linear equation in quasi-2-Banach spaces.Our results improve and generalize many results of literature.展开更多
This paper proposes the combined Laplace-Adomian decomposition method (LADM) for solution two dimensional linear mixed integral equations of type Volterra-Fredholm with Hilbert kernel. Comparison of the obtained resul...This paper proposes the combined Laplace-Adomian decomposition method (LADM) for solution two dimensional linear mixed integral equations of type Volterra-Fredholm with Hilbert kernel. Comparison of the obtained results with those obtained by the Toeplitz matrix method (TMM) demonstrates that the proposed technique is powerful and simple.展开更多
Over the last few years, there has been a significant increase in attention paid to fractional differential equations, given their wide array of applications in the fields of physics and engineering. The recent develo...Over the last few years, there has been a significant increase in attention paid to fractional differential equations, given their wide array of applications in the fields of physics and engineering. The recent development of using fractional telegraph equations as models in some fields (e.g., the thermal diffusion in fractal media) has heightened the importance of examining the method of solutions for such equations (both approximate and analytic). The present work is designed to serve as a valuable contribution to work in this field. The key objective of this work is to propose a general framework that can be used to guide quadratic spline functions in order to create a numerical method for obtaining an approximation solution using the linear space-fractional telegraph equation. Additionally, the Von Neumann method was employed to measure the stability of the analytical scheme, which showed that the proposed method is conditionally stable. What’s more, the proposal contains a numerical example that illustrates how the proposed method can be implemented practically, whilst the error estimates and numerical stability results are discussed in depth. The findings indicate that the proposed model is highly effective, convenient and accurate for solving the relevant problems and is suitable for use with approximate solutions acquired through the two-dimensional differential transform method that has been developed for linear partial differential equations with space- and time-fractional derivatives.展开更多
This paper proposes a novel distributed optimization algorithm with fractional order dynamics to solve linear algebraic equations.Firstly,the authors proposed“Consensus+Projection”flow with fractional order dynamics...This paper proposes a novel distributed optimization algorithm with fractional order dynamics to solve linear algebraic equations.Firstly,the authors proposed“Consensus+Projection”flow with fractional order dynamics,which has more design freedom and the potential to obtain a better convergent performance than that of conventional first order algorithms.Moreover,the authors prove that the proposed algorithm is convergent under certain iteration order and step-size.Furthermore,the authors develop iteration order switching scheme with initial condition design to improve the convergence performance of the proposed algorithm.Finally,the authors illustrate the effectiveness of the proposed method with several numerical examples.展开更多
The paper is devoted to non-homogeneous second-order differential equations with polynomial right parts and polynomial coefficients.We derive estimates for the partial sums and products of the zeros of solutions to th...The paper is devoted to non-homogeneous second-order differential equations with polynomial right parts and polynomial coefficients.We derive estimates for the partial sums and products of the zeros of solutions to the considered equations.These estimates give us bounds for the function counting the zeros of solutions and information about the zero-free domains.展开更多
Continuously differentiable radial basis functions (C<sup>∞</sup>-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are ...Continuously differentiable radial basis functions (C<sup>∞</sup>-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are full and can become very ill- conditioned. Similarly, the Hilbert and Vandermonde have full matrices and become ill-conditioned. The difference between a coefficient matrix generated by C<sup>∞</sup>-RBFs for partial differential or integral equations and Hilbert and Vandermonde systems is that C<sup>∞</sup>-RBFs are very sensitive to small changes in the adjustable parameters. These parameters affect the condition number and solution accuracy. The error terrain has many local and global maxima and minima. To find stable and accurate numerical solutions for full linear equation systems, this study proposes a hybrid combination of block Gaussian elimination (BGE) combined with arbitrary precision arithmetic (APA) to minimize the accumulation of rounding errors. In the future, this algorithm can execute faster using preconditioners and implemented on massively parallel computers.展开更多
Background:Early singular nodular hepatocellular carcinoma(HCC)is an ideal surgical indication in clinical practice.However,almost half of the patients have tumor recurrence,and there is no reliable prognostic predict...Background:Early singular nodular hepatocellular carcinoma(HCC)is an ideal surgical indication in clinical practice.However,almost half of the patients have tumor recurrence,and there is no reliable prognostic prediction tool.Besides,it is unclear whether preoperative neoadjuvant therapy is necessary for patients with early singular nodular HCC and which patient needs it.It is critical to identify the patients with high risk of recurrence and to treat these patients preoperatively with neoadjuvant therapy and thus,to improve the outcomes of these patients.The present study aimed to develop two prognostic models to preoperatively predict the recurrence-free survival(RFS)and overall survival(OS)in patients with singular nodular HCC by integrating the clinical data and radiological features.Methods:We retrospective recruited 211 patients with singular nodular HCC from December 2009 to January 2019 at Eastern Hepatobiliary Surgery Hospital(EHBH).They all met the surgical indications and underwent radical resection.We randomly divided the patients into the training cohort(n=132)and the validation cohort(n=79).We established and validated multivariate Cox proportional hazard models by the preoperative clinicopathologic factors and radiological features for association with RFS and OS.By analyzing the receiver operating characteristic(ROC)curve,the discrimination accuracy of the models was compared with that of the traditional predictive models.Results:Our RFS model was based on HBV-DNA score,cirrhosis,tumor diameter and tumor capsule in imaging.RFS nomogram had fine calibration and discrimination capabilities,with a C-index of 0.74(95%CI:0.68-0.80).The OS nomogram,based on cirrhosis,tumor diameter and tumor capsule in imaging,had fine calibration and discrimination capabilities,with a C-index of 0.81(95%CI:0.74-0.87).The area under the receiver operating characteristic curve(AUC)of our model was larger than that of traditional liver cancer staging system,Korea model and Nomograms in Hepatectomy Patients with Hepatitis B VirusRelated Hepatocellular Carcinoma,indicating better discrimination capability.According to the models,we fitted the linear prediction equations.These results were validated in the validation cohort.Conclusions:Compared with previous radiography model,the new-developed predictive model was concise and applicable to predict the postoperative survival of patients with singular nodular HCC.Our models may preoperatively identify patients with high risk of recurrence.These patients may benefit from neoadjuvant therapy which may improve the patients’outcomes.展开更多
The representation for the Moore-Penrose inverse of the matrix[AC BD]is derived by using the solvability theory of linear equations,where A∈C^(m×n),B∈C^(m×p),C∈C^(q×n)and D∈C^(q×p),with which s...The representation for the Moore-Penrose inverse of the matrix[AC BD]is derived by using the solvability theory of linear equations,where A∈C^(m×n),B∈C^(m×p),C∈C^(q×n)and D∈C^(q×p),with which some special cases are discussed.展开更多
文摘Let K be a proper cone in R^x,let A be an n×n real matrix that satisfies AK(?)K,letb be a given vector of K,and let λbe a given positive real number.The following two lin-ear equations are considered in this paper:(i)(λⅠ_n-A)x=b,x∈K,and(ii)(A-λⅠ_n)x=b,x∈K.We obtain several equivalent conditions for the solvability of the first equation.
文摘In this paper,we study the multivariate linear equations with arbitrary positive integral coefficients.Under the Generalized Riemann Hypothesis,we obtained the asymptotic formula for the linear equations with more than five prime variables.This asymptotic formula is composed of three parts,that is,the first main term,the explicit second main term and the error term.Among them,the first main term is similar with the former one,the explicit second main term is relative to the non-trivial zeros of Dirichlet L-functions,and our error term improves the former one.
文摘In this paper, we consider solving dense linear equations on Dawning1000 byusing matrix partitioning technique. Based on this partitioning of matrix, we give aparallel block LU decomposition method. The efficiency of solving linear equationsby different ways is analysed. The numerical results are given on Dawning1000.By running our parallel program, the best speed up on 32 processors is over 25.
基金supported by the National Science Foundation of China(41275063 and 11401575)
文摘In this article, we will derive local elliptic type gradient estimates for positive solutions of linear parabolic equations(?-?/(?t))u(x, t) + h(x,t)u(x,t) = 0 and nonlinear parabolic equations(?-?-/(?t))u(x,t) + h(x, t)u^p(x,t) = 0(p > 1) on Riemannian manifolds.As applications, we obtain some theorems of Liouville type for positive ancient solutions of such equations. Our results generalize that of Souplet-Zhang([1], Bull. London Math. Soc.38(2006), 1045-1053) and the author([2], Nonlinear Anal. 74(2011), 5141-5146).
基金supported by the National Natural Science Foundation of China(Nos.90816024,10872017,and 10876100)the Defense Industrial Technology Development Program(Nos.A2120110001 and 2120110011)the 111 Project(No.B07009)
文摘Based on linear interval equations, an accurate interval finite element method for solving structural static problems with uncertain parameters in terms of optimization is discussed.On the premise of ensuring the consistency of solution sets, the original interval equations are equivalently transformed into some deterministic inequations.On this basis, calculating the structural displacement response with interval parameters is predigested to a number of deterministic linear optimization problems.The results are proved to be accurate to the interval governing equations.Finally, a numerical example is given to demonstrate the feasibility and efficiency of the proposed method.
文摘This paper deals with the Hyers-Ulam stability of the nonhomogeneous linear dynamic equation x~?(t)-ax(t) = f(t), where a ∈ R^+. The main results can be regarded as a supplement of the stability results of the corresponding homogeneous linear dynamic equation obtained by Anderson and Onitsuka(Anderson D R, Onitsuka M. Hyers-Ulam stability of first-order homogeneous linear dynamic equations on time scales. Demonstratio Math., 2018, 51: 198–210).
文摘Integral equations theoretical parts and applications have been studied and investigated in previous works. In this work, results on investigations of the uniqueness of the Fredholm-Stiltjes linear integral equations solutions of the third kind were considered. Volterra integral equations of the first and third kind with smooth kernels were studied, and proof of the existence of a multiparameter family of solutions is described. Additionally, linear Fredholm integral equations of the first kind were investigated, for which Lavrent’ev regularizing operators were constructed.
基金supported by the Natural Science Foundation of Hubei Province,China (2022CFB444)the Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (NUAA)+1 种基金supported by the NSFC (12031006)the Fundamental Research Funds for the Central Universities of China.
文摘In this work,we study the linearized Landau equation with soft potentials and show that the smooth solution to the Cauchy problem with initial datum in L^(2)(ℝ^(3))enjoys an analytic regularization effect,and that the evolution of the analytic radius is the same as the heat equations.
基金supported by the National Natural Science Foundation of China(Grant No.12031004 and Grant No.12271474,61877054)the Fundamental Research Foundation for the Central Universities(Project No.K20210337)+1 种基金the Zhejiang University Global Partnership Fund,188170+194452119/003partially funded by a state task of Russian Fundamental Investigations(State Registration No.FFSG-2024-0002)。
文摘Fundamental matrix operations and solving linear systems of equations are ubiquitous in scientific investigations.Using the‘sender-receiver’model,we propose quantum algorithms for matrix operations such as matrix-vector product,matrix-matrix product,the sum of two matrices,and the calculation of determinant and inverse matrix.We encode the matrix entries into the probability amplitudes of the pure initial states of senders.After applying proper unitary transformation to the complete quantum system,the desired result can be found in certain blocks of the receiver’s density matrix.These quantum protocols can be used as subroutines in other quantum schemes.Furthermore,we present an alternative quantum algorithm for solving linear systems of equations.
文摘Hessian matrices are square matrices consisting of all possible combinations of second partial derivatives of a scalar-valued initial function. As such, Hessian matrices may be treated as elementary matrix systems of linear second-order partial differential equations. This paper discusses the Hessian and its applications in optimization, and then proceeds to introduce and derive the notion of the Jaffa Transform, a new linear operator that directly maps a Hessian square matrix space to the initial corresponding scalar field in nth dimensional Euclidean space. The Jaffa Transform is examined, including the properties of the operator, the transform of notable matrices, and the existence of an inverse Jaffa Transform, which is, by definition, the Hessian matrix operator. The Laplace equation is then noted and investigated, particularly, the relation of the Laplace equation to Poisson’s equation, and the theoretical applications and correlations of harmonic functions to Hessian matrices. The paper concludes by introducing and explicating the Jaffa Theorem, a principle that declares the existence of harmonic Jaffa Transforms, which are, essentially, Jaffa Transform solutions to the Laplace partial differential equation.
基金supported by Shanghai Center for Mathematical Science China Scholarship Council(201206105015)the National Science Foundation of China(11171119,11001057,11571049)the Natural Science Foundation of Guangdong Province in China(2014A030313422)
文摘In this paper, we mainly investigate entire solutions of complex differential equations with coefficients involving exponential functions, and obtain the dynamical properties of the solutions, their derivatives and primitives. With some conditions on coefficients, for the solutions, their derivatives and their primitives, we consider the common limiting directions of Julia set and the existence of Baker wandering domain.
基金Project supported by the National Natural Science Foundation of China(Nos.11272183,11572176,11402167,11202147,and 11332007)the National Program on Key Basic Research Project of China(No.2014CB744801)
文摘It is widely accepted that a robust and efficient method to compute the linear spatial amplified rate ought to be developed in three-dimensional(3D) boundary layers to predict the transition with the e^N method,especially when the boundary layer varies significantly in the spanwise direction.The 3D-linear parabolized stability equation(3DLPSE) approach,a 3D extension of the two-dimensional LPSE(2D-LPSE),is developed with a plane-marching procedure for investigating the instability of a 3D boundary layer with a significant spanwise variation.The method is suitable for a full Mach number region,and is validated by computing the unstable modes in 2D and 3D boundary layers,in both global and local instability problems.The predictions are in better agreement with the ones of the direct numerical simulation(DNS) rather than a 2D-eigenvalue problem(EVP) procedure.These results suggest that the plane-marching 3D-LPSE approach is a robust,efficient,and accurate choice for the local and global instability analysis in 2D and 3D boundary layers for all free-stream Mach numbers.
基金AISTDF,DST India for the research grant vide project No.CRD/2018/000017。
文摘In this paper,we study the hyperstability for the general linear equation f(ax+by)=Af(x)+Bf(y)in the setting of complete quasi-2-Banach spaces.We first extend the main fixed point result of Brzdek and Ciepliński(Acta Mathematica Scientia,2018,38 B(2):377-390)to quasi-2-Banach spaces by defining an equivalent quasi-2-Banach space.Then we use this result to generalize the main results on the hyperstability for the general linear equation in quasi-2-Banach spaces.Our results improve and generalize many results of literature.
文摘This paper proposes the combined Laplace-Adomian decomposition method (LADM) for solution two dimensional linear mixed integral equations of type Volterra-Fredholm with Hilbert kernel. Comparison of the obtained results with those obtained by the Toeplitz matrix method (TMM) demonstrates that the proposed technique is powerful and simple.
文摘Over the last few years, there has been a significant increase in attention paid to fractional differential equations, given their wide array of applications in the fields of physics and engineering. The recent development of using fractional telegraph equations as models in some fields (e.g., the thermal diffusion in fractal media) has heightened the importance of examining the method of solutions for such equations (both approximate and analytic). The present work is designed to serve as a valuable contribution to work in this field. The key objective of this work is to propose a general framework that can be used to guide quadratic spline functions in order to create a numerical method for obtaining an approximation solution using the linear space-fractional telegraph equation. Additionally, the Von Neumann method was employed to measure the stability of the analytical scheme, which showed that the proposed method is conditionally stable. What’s more, the proposal contains a numerical example that illustrates how the proposed method can be implemented practically, whilst the error estimates and numerical stability results are discussed in depth. The findings indicate that the proposed model is highly effective, convenient and accurate for solving the relevant problems and is suitable for use with approximate solutions acquired through the two-dimensional differential transform method that has been developed for linear partial differential equations with space- and time-fractional derivatives.
基金supported by the National Natural Science Foundation of China under Grant Nos.62103003,62073001,and 61973002the Anhui Provincial Key Research and Development Project under Grant2022i01020013+3 种基金the University Synergy Innovation Program of Anhui Province under Grant No.GXXT-2021-010the Anhui Provincial Natural Science Foundation under Grant No.2008085J32the National Postdoctoral Program for Innovative Talents under Grant No.BX20180346the General Financial Grant from the China Postdoctoral Science Foundation under Grant No.2019M660834。
文摘This paper proposes a novel distributed optimization algorithm with fractional order dynamics to solve linear algebraic equations.Firstly,the authors proposed“Consensus+Projection”flow with fractional order dynamics,which has more design freedom and the potential to obtain a better convergent performance than that of conventional first order algorithms.Moreover,the authors prove that the proposed algorithm is convergent under certain iteration order and step-size.Furthermore,the authors develop iteration order switching scheme with initial condition design to improve the convergence performance of the proposed algorithm.Finally,the authors illustrate the effectiveness of the proposed method with several numerical examples.
文摘The paper is devoted to non-homogeneous second-order differential equations with polynomial right parts and polynomial coefficients.We derive estimates for the partial sums and products of the zeros of solutions to the considered equations.These estimates give us bounds for the function counting the zeros of solutions and information about the zero-free domains.
文摘Continuously differentiable radial basis functions (C<sup>∞</sup>-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are full and can become very ill- conditioned. Similarly, the Hilbert and Vandermonde have full matrices and become ill-conditioned. The difference between a coefficient matrix generated by C<sup>∞</sup>-RBFs for partial differential or integral equations and Hilbert and Vandermonde systems is that C<sup>∞</sup>-RBFs are very sensitive to small changes in the adjustable parameters. These parameters affect the condition number and solution accuracy. The error terrain has many local and global maxima and minima. To find stable and accurate numerical solutions for full linear equation systems, this study proposes a hybrid combination of block Gaussian elimination (BGE) combined with arbitrary precision arithmetic (APA) to minimize the accumulation of rounding errors. In the future, this algorithm can execute faster using preconditioners and implemented on massively parallel computers.
基金supported by grants from the Shanghai Rising-Star Program(19QA1408700)the National Natural Science Founda-tion of China(81972575 and 81521091)Clinical Research Plan of SHDC(SHDC2020CR5007)。
文摘Background:Early singular nodular hepatocellular carcinoma(HCC)is an ideal surgical indication in clinical practice.However,almost half of the patients have tumor recurrence,and there is no reliable prognostic prediction tool.Besides,it is unclear whether preoperative neoadjuvant therapy is necessary for patients with early singular nodular HCC and which patient needs it.It is critical to identify the patients with high risk of recurrence and to treat these patients preoperatively with neoadjuvant therapy and thus,to improve the outcomes of these patients.The present study aimed to develop two prognostic models to preoperatively predict the recurrence-free survival(RFS)and overall survival(OS)in patients with singular nodular HCC by integrating the clinical data and radiological features.Methods:We retrospective recruited 211 patients with singular nodular HCC from December 2009 to January 2019 at Eastern Hepatobiliary Surgery Hospital(EHBH).They all met the surgical indications and underwent radical resection.We randomly divided the patients into the training cohort(n=132)and the validation cohort(n=79).We established and validated multivariate Cox proportional hazard models by the preoperative clinicopathologic factors and radiological features for association with RFS and OS.By analyzing the receiver operating characteristic(ROC)curve,the discrimination accuracy of the models was compared with that of the traditional predictive models.Results:Our RFS model was based on HBV-DNA score,cirrhosis,tumor diameter and tumor capsule in imaging.RFS nomogram had fine calibration and discrimination capabilities,with a C-index of 0.74(95%CI:0.68-0.80).The OS nomogram,based on cirrhosis,tumor diameter and tumor capsule in imaging,had fine calibration and discrimination capabilities,with a C-index of 0.81(95%CI:0.74-0.87).The area under the receiver operating characteristic curve(AUC)of our model was larger than that of traditional liver cancer staging system,Korea model and Nomograms in Hepatectomy Patients with Hepatitis B VirusRelated Hepatocellular Carcinoma,indicating better discrimination capability.According to the models,we fitted the linear prediction equations.These results were validated in the validation cohort.Conclusions:Compared with previous radiography model,the new-developed predictive model was concise and applicable to predict the postoperative survival of patients with singular nodular HCC.Our models may preoperatively identify patients with high risk of recurrence.These patients may benefit from neoadjuvant therapy which may improve the patients’outcomes.
文摘The representation for the Moore-Penrose inverse of the matrix[AC BD]is derived by using the solvability theory of linear equations,where A∈C^(m×n),B∈C^(m×p),C∈C^(q×n)and D∈C^(q×p),with which some special cases are discussed.
